A055025 -id:A055025 - cp's OEIS frontend

A344124 Decimal expansion of Sum_{i > 0} 1/A001481(i)^3.

1, 1, 5, 4, 5, 3, 8, 3, 3, 0, 4, 7, 6, 3, 8, 8, 9, 4, 3, 9, 2, 2, 1, 0, 6, 5, 9, 4, 5, 5, 5, 5, 1, 6, 8, 2, 9, 8, 9, 8, 7, 7, 5, 1, 9, 7, 4, 4, 8, 7

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an analog of zeta(3) (= Apéry's constant = A002117), where Euler's zeta(3) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

			1.1545383304763889439221065945555168298987751974487...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025.

Cf. A344123, A344125.

Equals Sum_{i > 0} 1/A001481(i)^3.
Equals Product_{i > 0} 1/(1-A055025(i)^-3).
Equals 1/(1-prime(1)^(-3)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-3)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-6)), where prime(n) = A000040(n).
Equals zeta_{2,0} (3) * zeta_{4,1} (3) * zeta_{4,3} (6), where zeta_{2,0} (s) = 2^s/(2^s - 1).

A344125 Decimal expansion of Sum_{i > 0} 1/A001481(i)^4.

Original entry on oeis.org

1, 0, 6, 8, 5, 9, 2, 1, 0, 5, 6, 5, 4, 9, 9, 0, 1, 3, 5, 2, 0, 2, 9, 4, 8, 0, 2, 0, 7, 4, 3, 2, 4, 3, 6, 1, 3, 6, 1, 3, 3, 3, 5, 9, 0, 8, 1, 0, 1, 7

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an analog of zeta(4) (= Pi^4/90 = A013662), where Euler's zeta(4) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

			1.0685921056549901352029480207432436136133359081017...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025.

Cf. A344123, A344124.

Equals Sum_{i > 0} 1/A001481(i)^4.
Equals Product_{i > 0} 1/(1-A055025(i)^-4).
Equals 1/(1-prime(1)^(-4)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-4)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-8)), where prime(n) = A000040(n).
Equals zeta_{2,0} (4) * zeta_{4,1} (4) * zeta_{4,3} (8), where zeta_{2,0} (s) = 2^s/(2^s - 1).

A364868 Numbers k such that 4*k+1 is the norm of a Gaussian prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 10, 12, 13, 15, 18, 22, 24, 25, 27, 28, 30, 34, 37, 39, 43, 45, 48, 49, 57, 58, 60, 64, 67, 69, 70, 73, 78, 79, 84, 87, 88, 90, 93, 97, 99, 100, 102, 105, 108, 112, 114, 115, 127, 130, 132, 135, 139, 142, 144, 148, 150, 153, 154, 160, 163, 165, 168, 169

Offset: 1

Jianing Song, Aug 11 2023

Numbers k such that 4*k+1 is a prime or the square of a prime congruent to 3 modulo 4.

If p is a Gaussian prime of norm 4*a(n)+1 (there are two up to association if a(n) is a prime, one if a(n) is the square of a prime), then for any Gaussian integer x, we have x^a(n) == 0, 1, i, -1 or -i (mod p) where i is a primitive fourth root of unity.

			2 is a term since 4*2+1 is the norm of the Gaussian prime 3.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A055025, A364869.

Contains 6*A024702 as a subsequence.

isA364868(n) = isA055025(4*n+1) \\ See A055025 for its program

a(n) = (A055025(n+1) - 1)/4.

A185271 Differences between consecutive norms of Gaussian primes.

Original entry on oeis.org

3, 4, 4, 4, 12, 8, 4, 8, 4, 8, 12, 16, 8, 4, 8, 4, 8, 16, 12, 8, 16, 8, 12, 4, 32, 4, 8, 16, 12, 8, 4, 12, 20, 4, 20, 12, 4, 8, 12, 16, 8, 4, 8, 12, 12, 16, 8, 4, 48, 12, 8, 12, 16, 12, 8, 16, 8, 12, 4, 24, 12, 8, 12, 4, 24, 8, 24, 24, 4, 8, 4, 24, 12, 12, 8

Offset: 1

Patrick P Sheehan, Jan 25 2012

If this sequence goes to infinity fast enough then the Gaussian moat-crossing problem is solved and it is impossible to walk to infinity in the complex plane using steps of bounded length stepping only on Gaussian primes.

			The first Gaussian prime (restricting ourselves to the first octant) is 1+i which has norm 2 (1^2+1^2). The second is 2+i with norm 5 (2^2+1^2). The difference in those norms is 3, the first term in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric W. Weisstein, MathWorld: Gaussian Prime
Eric W. Weisstein, MathWorld: Moat-Crossing Problem
Wikipedia, Gaussian Integer
Index entries for Gaussian integers and primes

Cf. A055025 (norms of Gaussian primes).

a(n) = A055025(n+1) - A055025(n).

A269790 Primes p such that 2*p + 79 is a square.

Original entry on oeis.org

73, 181, 2341, 4861, 6121, 9901, 12601, 18973, 20161, 26641, 47701, 51481, 59473, 61561, 68041, 79561, 81973, 84421, 94573, 110881, 157321, 185401, 192781, 207973, 231841, 244261, 248473, 270073, 292573, 335341, 365473, 440821, 446473, 452161, 475273

Offset: 1

Vincenzo Librandi, Mar 24 2016

nonn

Primes of the form 2*k^2 + 2*k - 39.

2*p + h is not verified if h is an odd prime that belongs to A055025 because (2*h-1)/2 is a multiple of 2.

			a(1) = 73 because 2*73 + 79 = 225, which is a square.

Cf. A000040.
Subsequence of A002144, A045433, A061237, A068228.
Cf. similar sequences listed in A269784.

[p: p in PrimesUpTo(600000) | IsSquare(2*p+79)];

Select[Prime[Range[50000]], IntegerQ[Sqrt[2 # + 79]] &]

lista(nn) = {forprime(p=2, nn, if(issquare(2*p + 79), print1(p, ", "))); } \\ Altug Alkan, Mar 24 2016

from sympy import isprime
from gmpy2 import is_square
for p in range(0,1000000):
    if(is_square(2*p+79) and isprime(p)):print(p)
# Soumil Mandal, Apr 03 2016

A333597 The number of unit cells intersected by the circumference of a circle centered on the origin with radius squared equal to the norm of the Gaussian integers A001481(n).

Original entry on oeis.org

0, 4, 8, 12, 12, 16, 20, 20, 20, 28, 28, 32, 28, 28, 36, 36, 40, 36, 44, 44, 44, 44, 44, 52, 48, 52, 52, 52, 52, 60, 52, 60, 64, 60, 60, 60, 68, 68, 60, 68, 68, 68, 72, 68, 76, 76, 76, 76, 76, 76, 76, 84, 84, 76, 88, 76, 84, 84, 92, 84, 92

Offset: 1

Scott R. Shannon, Mar 28 2020

nonn

Draw a circle on a 2D square grid centered at the origin with a radius squared equal to the norm of the Gaussian integers A001481(n). See the images in the links. This sequence gives the number of unit cells intersected by the circumference of the circle. Equivalently this is the number of intersections of the circumference with the x and y integer grid lines.

Scott R. Shannon, Illustration for n = 3. The circle has a radius squared of 2, resulting in 8 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 4. The circle has a radius squared of 4, resulting in 12 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 8. The circle has a radius squared of 10, resulting in 20 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 12. The circle has a radius squared of 18, resulting in 32 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 13. The circle has a radius squared of 20, resulting in 28 unit cells intersected/intersection points. This is the first term where the number of intersection points decreases relative to the previous term.
Scott R. Shannon, Illustration for n = 26. The circle has a radius squared of 52, resulting in 52 unit cells intersected/intersection points.
Wikipedia, Gaussian integer.

Cf. A001481, A055025, A057655, A119439, A242118 (a subsequence of this sequence), A234300.

a(n) = 4*A234300(2*(n-1)). - Andrey Zabolotskiy, Feb 22 2025

A120119 Multiplicative function from integers to sums of two squares.

Original entry on oeis.org

1, 2, 5, 4, 9, 10, 13, 8, 25, 18, 17, 20, 29, 26, 45, 16, 37, 50, 41, 36, 65, 34, 49, 40, 81, 58, 125, 52, 53, 90, 61, 32, 85, 74, 117, 100, 73, 82, 145, 72, 89, 130, 97, 68, 225, 98, 101, 80, 169, 162, 185, 116, 109, 250, 153, 104, 205, 106, 113, 180, 121, 122, 325, 64

Offset: 1

Franklin T. Adams-Watters, Aug 15 2006

The values here are a rearrangement of A001481. The numbers in A001481 have unique factorization (with factors restricted to A001481), with the numbers in A055025 being the "primes".

Cf. A001481, A055025, A000040.

Totally multiplicative with a(Prime(k)) = A055025(k).

A385493 Number of distinct states in Conway's Game of Life acting on a (2n+1) X (2n+1) toroidal grid starting with (x,y) turned on if and only if x-n + (y-n)*i is a Gaussian prime.

Original entry on oeis.org

1, 1, 1, 6, 9, 4, 4, 5, 14, 12, 17, 5, 8, 19, 15, 34, 20, 21, 19, 77, 52, 29, 58, 39, 27, 27, 68, 31, 27, 27, 27, 70, 49, 129, 83, 43, 153, 40, 82, 128, 60, 457, 436, 79, 99, 71, 71, 178, 125, 281, 121, 121, 94, 231, 94, 94, 385, 122, 94, 94, 175, 306, 156

Offset: 0

Luke Bennet, Jun 30 2025

nonn

			For a(3), the sequence of Conway's Game of Life is
 | . o . o . o . | . o . o . o . | . o . . . o . |
 | o . o . o . o | o . . . . . o | o o o o o o o |
 | . o o . o o . | . . o . o . . | . o . . . o . |
 | o . . . . . o | o . . . . . o | . o . . . o . |
 | . o o . o o . | . . o . o . . | . o . . . o . |
 | o . o . o . o | o . . . . . o | o o o o o o o |
 | . o . o . o . | . o . o . o . | . o . . . o . |
  (generation 1)  (generation 2)  (generation 3)
 | . . . o . . . | . . o o o . . | . o . . . o . |
 | . . . o . . . | . . o o o . . | o . . . . . o |
 | . . . o . . . | o o . o . o o | . . . . . . . |
 | o o o . o o o | o o o . o o o | . . . . . . . |
 | . . . o . . . | o o . o . o o | . . . . . . . |
 | . . . o . . . | . . o o o . . | o . . . . . o |
 | . . . o . . . | . . o o o . . | . o . . . o . |
  (generation 4)  (generation 5)  (generation 6)
Every generation after 6 is identical to generation 6, so this sequence has 6 unique states. Thus, a(3) = 6.

Cf. A055025.

import torch
import numpy as np
def prime_mask(limit):
    is_prime = torch.ones(limit + 1, dtype=torch.bool)
    is_prime[:2] = False
    for i in range(2, int(limit**0.5) + 1):
        if is_prime[i]:
            is_prime[i*i : limit+1 : i] = False
    return is_prime
def Gauss_primes(N):
    A, B = torch.meshgrid(torch.arange(-N, N+1), torch.arange(-N, N+1))
    norm = A**2 + B**2
    is_prime = prime_mask(2*N**2)
    mask = (A != 0) & (B != 0) & is_prime[norm]
    axis_mask = ((A == 0) ^ (B == 0))
    axis_val = (A + B).abs()
    axis_mask &= is_prime[axis_val] & ((axis_val % 4) == 3)
    return mask | axis_mask
def update(G):
    shifts = [(1,0),(1,1),(0,1),(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)]
    neighbors = sum(torch.roll(G, shifts=shift, dims=(0,1)) for shift in shifts)
    return (G & ((neighbors == 2) | (neighbors == 3))) | (~G & (neighbors == 3))
def a(n):
    if n == 0 or n == 1:
        return 1
    G = Gauss_primes(n).to("cuda").to(torch.uint8)
    seen, step = set(), 0
    while True:
        flat = G.flatten().to("cpu").numpy()
        key = bytes(np.packbits(flat))
        if key in seen:
            return step
        seen.add(key)
        G = update(G)
        step += 1

cp's OEIS Frontend

A344124 Decimal expansion of Sum_{i > 0} 1/A001481(i)^3.

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A344125 Decimal expansion of Sum_{i > 0} 1/A001481(i)^4.

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A364868 Numbers k such that 4*k+1 is the norm of a Gaussian prime.

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PARI

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A185271 Differences between consecutive norms of Gaussian primes.

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A269790 Primes p such that 2*p + 79 is a square.

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Magma

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A333597 The number of unit cells intersected by the circumference of a circle centered on the origin with radius squared equal to the norm of the Gaussian integers A001481(n).

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A120119 Multiplicative function from integers to sums of two squares.

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A385493 Number of distinct states in Conway's Game of Life acting on a (2n+1) X (2n+1) toroidal grid starting with (x,y) turned on if and only if x-n + (y-n)*i is a Gaussian prime.

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Python